The picture below is Theorem 2.10 (Linear functionals separate of $L^p$) in page 56 of Lieb's "Analysis" book.
Question: How could I understand that the restriction of sigma-finite can be lifted by invoking transfinite induction in the case of $L^\infty$?
The phrasing "by invoking transfinite induction" is indeed rather cryptic here. In the case $p=\infty$, there is no obvious way to use transfinite induction to modify the given argument to work for non-$\sigma$-finite $\Omega$.
I would guess that the book is instead attempting to allude to the Hahn-Banach theorem, which implies that the statement of the theorem is valid with $L^p(\Omega)$ replaced by any normed vector space at all. The Hahn-Banach theorem can be proven by transfinite induction, though the details are a bit complicated. In the context of proving Theorem 2.10 for general normed vector spaces, the idea is to start with the functional defined just on the span of $f$ that sends $f$ to $\|f\|$, and then by transfinite induction extend it to larger and larger linear subspaces of your normed vector space while keeping the functional still bounded. The tricky part is showing that it always is possible to extend the functional to one more vector while keeping it still bounded (with the same norm).